Appendix: Mean field solutions

ATOM INTERFEROMETRY: LINEAR QUANTUM MEASUREMENT THEORY AND THE PRECISION LIMIT

5.10 Appendix: Mean field solutions

Typically, in an interferometric process, the light-atom interaction time is very short compared to the free evolution time of the atom cloud, and the center-of- mass velocity of the atom cloud is very low, typically∼ 2 cm/s. Therefore to the leading order, we can treat the atom’s center-of-mass motion to be static during the interaction process, that is,𝑣_𝐴 ≈ 𝑣_𝐵 ≈ 0. We also note that the spatial size of optical fields are much larger than the size of the atom cloud, therefore we can approximate the mean value of the optical fields to be almost constants during the interaction process.

The zeroth-order of the equations of motion is simple:

𝜕_𝑡𝜙¯⁺

𝐴= (𝑔_𝐴𝜙¯⁺

𝑝𝜙¯⁻

𝑐)𝜙¯⁺

𝐵, (𝜕_𝑡+𝜕_𝑧)𝜙¯⁺

𝑐 =0,

𝜕_𝑡𝜙¯⁺

𝐵 = (𝑔_𝐵𝜙¯⁻

𝑝𝜙¯⁺

𝑐)𝜙¯⁺

𝐴, (𝜕_𝑡−𝜕_𝑧)𝜙¯⁺

𝑝=0.

(5.119) We ignore the r.h.s. of the equation for the optical field because the photon number is much larger than the atom number. Since ¯𝜙_𝑝and ¯𝜙_𝑐are almost constant, therefore

we can rewrite the zeroth-order atom equations to be:

𝜕_𝑡𝜙¯⁺

𝐴 =−𝑖Ω𝑒^{𝑖 𝜑𝑝 𝑐}𝜙¯⁺

𝐵, 𝜕_𝑡𝜙¯⁺

𝐵 =−𝑖Ω𝑒^{−𝑖 𝜑𝑝 𝑐}𝜙¯⁺

𝐴, (5.120)

where 𝜑_{𝑝 𝑐} is the phase difference between the control field and passive field,Ω is the Ramsey frequency. Here, we make use of the approximation 𝑔_𝐴 = 𝑔_𝐵 := 𝑔_𝑎 therebyΩ =|𝑔_𝑎𝜙¯_𝑝𝜙¯_𝑐|. The full solution of this equation is:

¯ 𝜙⁺

𝐴(𝑡) =𝜙¯⁺

𝐴(0)cosΩ𝑡−𝑖𝜙¯⁺

𝐵(0)𝑒^{𝑖 𝜑𝑐 𝑝}sinΩ𝑡 ,

¯ 𝜙⁺

𝐵(𝑡) =𝜙¯⁺

𝐵(0)cosΩ𝑡−𝑖𝜙¯⁺

𝐴(0)𝑒^{−𝑖 𝜑𝑐 𝑝}sinΩ𝑡 .

(5.121) The gravitational wave signal will be carried by the𝜑_{𝑐 𝑝}. Clearly, this is where the matrixM(𝜃 , 𝜑)in the main text comes from.

References

[1] LIGO Scientific Collaboration and Virgo Collaboration. Observation of grav- itational waves from a binary black hole merger.Phys. Rev. Lett, 116:061102, Feb. 2016.

[2] LIGO Scientific Collaboration and Virgo Collaboration. Gw170817: Obser- vation of gravitational waves from a binary neutron star inspiral. Phys. Rev.

Lett, 119:161101, Oct 2017.

[3] Gregory M Harry and the LIGO Scientific Collaboration. Advanced ligo:

the next generation of gravitational wave detectors. Classical and Quantum Gravity, 27:084006, 2010.

[4] Virgo Collaboration. Advanced virgo: a second-generation interferometric gravitational wave detector.Classical and Quantum Gravity, 32:024001, 2015.

[5] KAGRA Collaboration. Kagra: 2.5 generation interferometric gravitational wave detector. Nature Astronomy, 3:35–40, Jan 2019.

[6] S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran.

Gravitational wave detection with atom interferometry. Physics Letter B, 678:37–40, jul 2009.

[7] Savas Dimopoulos, Peter W. Graham, Jason M. Hogan, Mark A. Kasevich, and Surjeet Rajendran. Atomic gravitational wave interferometric sensor. Phys.

Rev. D, 78:122002, Dec 2008.

[8] G M Tino and F Vetrano. Is it possible to detect gravitational waves with atom interferometers? Classical and Quantum Gravity, 24(9):2167–2178, apr 2007.

[9] N. Yu and M. Tinto. Gravitational wave detection with single-laser atom interferometers. Gen. Rel. Grav., 43:1943–1952, jul 2011.

[10] P. L. Bender. Limitations of atom interferometry for gravitational wave obser- vations in space. Gen. Relativ. Gravit., 44(711), Nov 2012.

[11] Peter W. Graham, Jason M. Hogan, Mark A. Kasevich, and Surjeet Rajendran.

New method for gravitational wave detection with atomic sensors. Phys. Rev.

Lett., 110:171102, apr 2013.

[12] Jan Harms, Bram J. J. Slagmolen, Rana X. Adhikari, M. Coleman Miller, Matthew Evans, Yanbei Chen, Holger Mueller, and Masaki Ando. Low- frequency terrestrial gravitational-wave detectors. Phys. Rev. D, 88:122003, Dec 2013.

[13] Remi Geiger. Future gravitational wave detectors based on atom interferometry.

In Gerard Auger and Eric Plagnol, editors, An Overview of Gravitational Waves: Theory, Sources and Detection, pages 285–313. World Scientific, 2017.

[14] Peter W. Graham and Sunghoon Jung. Localizing gravitational wave sources with single-baseline atom interferometers.Phys. Rev. D, 97:024052, Jan 2018.

[15] J. Junca, A. Bertoldi, D. O. Sabulsky, G. Lefevre, X. Zou, J.-B. Decitre, R. Geiger, A. Landragin, S. Gaffet, P. Bouyer, and B. Canuel. Characterizing earth gravity field fluctuations with the miga antenna for future gravitational wave detectors. Phys. Rev. D, 99:104026, may 2019.

[16] Ming-Sheng Zhan, Jin Wang, Wei-Tou Ni, Dong-Feng Gao, Gang Wang, Ling- Xiang He, Run-Bing Li, Lin Zhou, Xi Chen, Jia-Qi Zhong, Biao Tang, Zhan- Wei Yao, Lei Zhu, Zong-Yuan Xiong, Si-Bin Lu, Geng-Hua Yu, Qun-Feng Cheng, Min Liu, Yu-Rong Liang, Peng Xu, Xiao-Dong He, Min Ke, Zheng Tan, and Jun Luo. ZAIGA: Zhaoshan Long-baseline Atom Interferometer Gravitation Antenna. arXiv e-prints, page arXiv:1903.09288, Mar 2019.

[17] I. I. Rabi, J. R. Zacharias, S. Millman, and P. Kusch. A new method of measuring nuclear magnetic moment. Phys. Rev., 53:318–318, Feb. 1938.

[18] Norman F. Ramsey. A molecular beam resonance method with separated oscillating fields. Phys. Rev., 78:695–699, Jun 1950.

[19] Alexander D. Cronin, Jorg Schmiedmayer, and David E. Pritchard. Optics and interferometry with atoms and molecules.Rev. Mod. Phys, 81:1051–1129, Jul 2009.

[20] Savas Dimopoulos, Peter W. Graham, Jason M. Hogan, and Mark A. Kasevich.

Testing general relativity with atom interferometry.Phys. Rev. Lett, 98:111102, Mar 2007.

[21] Holger Mueller, Sheng-wey Chiow, Sven Herrmann, Steven Chu, and Keng- Yeow Chung. Atom-interferometry tests of the isotropy of post-newtonian

gravity. Phys. Rev. Lett., 100:031101, Jan 2008.

[22] Asimina Arvanitaki, Savas Dimopoulos, Andrew A. Geraci, Jason Hogan, and Mark Kasevich. How to test atom and neutron neutrality with atom interferometry. Phys. Rev. Lett., 100:120407, Mar 2008.

[23] Savas Dimopoulos, Peter W. Graham, Jason M. Hogan, and Mark A. Kasevich.

General relativistic effects in atom interferometry. Phys. Rev. D, 78:042003, Aug 2008.

[24] Keng-Yeow Chung, Sheng-Wey Chiow, Sven Herrmann, Steven Chu, and Holge Mueller. Atom interferometry tests of local lorentz invariance in gravity and electrodynamics. Phys. Rev. D, 80:016002, Jul 2009.

[25] Rym Bouchendira, Pierre Clade, Saida Guellati-Khelifa, Francois Nez, and Francois Biraben. New determination of the fine structure constant and test of the quantum electrodynamics. Phys. Rev. Lett, 106:080801, Feb. 2011.

[26] Michael A. Hohensee, Steven Chu, Achim Peters, and Holger Mueller. Equiv- alence principle and gravitational redshift. Phys. Rev. Lett, 106:151102, Apr 2011.

[27] Zhong-Kun Hu, Bu-Liang Sun, Xiao-Chun Duan, Min-Kang Zhou, Le-Le Chen, Su Zhan, Qiao-Zhen Zhang, and Jun Luo. Demonstration of an ultrahigh- sensitivity atom-interferometry absolute gravimeter.Phys. Rev. A, 88:043610, Oct 2013.

[28] G. Rosi, F. Sorrentino, L. Cacciapuoti, M. Prevedelli, and G. M. Tino. Precision measurement of the newtonian gravitational constant using cold atoms.Nature, 510:518 EP –, 06 2014.

[29] P. Hamilton, M. Jaffe, P. Haslinger, Q. Simmons, H. Müller, and J. Khoury.

Atom-interferometry constraints on dark energy.Science, 349:849–851, 2015.

[30] Xiao-Chun Duan, Xiao-Bing Deng, Min-Kang Zhou, Ke Zhang, Wen-Jie Xu, Feng Xiong, Yao-Yao Xu, Cheng-Gang Shao, Jun Luo, and Zhong-Kun Hu.

Test of the universality of free fall with atoms in different spin orientations.

Phys. Rev. Lett, 117:023001, Jul 2016.

[31] Benjamin Elder, Justin Khoury, Philipp Haslinger, Matt Jaffe, Holger Müller, and Paul Hamilton. Chameleon dark energy and atom interferometry. Phys.

Rev. D, 94:044051, Aug 2016.

[32] R. Parker, C. Yu, W. Zhong, J. Kwan, B. Estey, and H. Muller. Measurement of the fine structure constant as a test of the standard model.Science, 360:191–

195, 2018.

[33] Holger Mueller, Sheng-wey Chiow, Quan Long, Sven Herrmann, and Steven Chu. Atom interferometry with up to 24-photon-momentum-transfer beam splitters. Phys. Rev. Lett., 100:180405, May 2008.

[34] Pierre Clade, Saidada Guellati-Khelifa, Francois Nez, and Francois Biraben.

Large momentum beam splitter using bloch oscillations. Phys. Rev. Lett., 102:240402, Jun 2009.

[35] Onur Hosten, Nils J. Engelsen, Rajiv Krishnakumar, and Mark A. Kasevich.

Measurement noise 100 times lower than the quantum-projection limit using entangled atoms. Nature, 529:505 EP –, 01 2016.

[36] Paul Hamilton, Matt Jaffe, Justin M. Brown, Lothar Maisenbacher, Brian Estey, and Holger Mueller. Atom interferometry in an optical cavity.Phys. Rev. Lett., 114:100405, Mar 2015.

[37] B. Canuel, A. Bertoldi, L. Amand, E. Pozzo di Borgo, T. Chantrait, C. Dan- quigny, M. Dovale Álvarez, B. Fang, A. Freise, R. Geiger, J. Gillot, S. Henry, J. Hinderer, D. Holleville, J. Junca, G. Lefèvre, M. Merzougui, N. Mielec, T. Monfret, S. Pelisson, M. Prevedelli, S. Reynaud, I. Riou, Y. Rogister, S. Rosat, E. Cormier, A. Landragin, W. Chaibi, S. Gaffet, and P. Bouyer.

Exploring gravity with the miga large scale atom interferometer. Scientific Reports, 8(1):14064, 2018.

[38] V. B. Braginsky, F. Y. Khalili, and K. S. Thorne. Quantum Measurement.

Cambridge University Press, sep 1992.

[39] H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, and Sergey P.

Vyatchanin. Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics. Phys. Rev. D, 65:022002, Dec 2001.

[40] Alessandra Buonanno and Yanbei Chen. Signal recycled laser-interferometer gravitational-wave detectors as optical springs. Phys. Rev. D, 65:042001, Jan 2002.

[41] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland. Quantum projection noise:

Population fluctuations in two-level systems. Phys. Rev. A, 47:3554–3570, May 1993.

[42] J. Jacobson, G. Bjork, and Y. Yamamoto. Quantum limit for the atom-light interferometer. Applied Physics B: Lasers and Optics, 60:187–191, feb 1995.

[43] Jonathan P. Dowling. Correlated input-port, matter-wave interferometer:

Quantum-noise limits to the atom-laser gyroscope. Phys. Rev. A, 57:4736–

4746, jun 1998.

[44] Marlan O. Scully and Jonathan P. Dowling. Quantum-noise limits to matter- wave interferometry. Phys. Rev. A, 48:3186–3190, Oct 1993.

[45] Vladimir B. Braginsky, Mikhail L. Gorodetsky, Farid Ya. Khalili, Andrey B.

Matsko, Kip S. Thorne, and Sergey P. Vyatchanin. Noise in gravitational-wave detectors and other classical-force measurements is not influenced by test-mass quantization. Phys. Rev. D, 67:082001, Apr 2003.

[46] Haixing Miao, Rana X Adhikari, Yiqiu Ma, Belinda Pang, and Yanbei Chen.

Towards the fundamental quantum limit of linear measurements of classical signals. Phys. Rev. Lett., 119:050801, Aug 2017.

[47] John G. Baker and J. I. Thorpe. Comparison of atom interferometers and light interferometers as space-based gravitational wave detectors. Phys. Rev. Lett., 108:211101, may 2012.

[48] Christian J. Bordé. Theoretical tools for atom optics and interferometry.

Comptes Rendus de l’Académie des Sciences - Series IV - Physics, 2(2):509–

530, 2001.

[49] L. D. Landau and E. M. Lifshitz. Quantum mechanics, Course of Theoretical Physics. Oxford: Pergamon Press, 1965.

C h a p t e r 6

HIGH-PRECISION MODELING FOR GRAVITATIONAL WAVES